Related papers: Advanced Mechanics. Mathematical Introduction
We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis.…
Dynamic equations of non-relativistic mechanics are written in covariant-coordinate form in terms of relative velocities and accelerations with respect to an arbitrary reference frame. The notions of the non-relativistic reference frame,…
Absolute space is eliminated from the body of mechanics by gauging translations and rotations in the Lagrangian of a classical system. The procedure implies the addition of compensating terms to the kinetic energy, in such a way that the…
From the principle that there is no absolute description of a physical state, we advance the approach according to which one should be able to describe the physics from the perspective of a quantum particle. The kinematics seen from this…
We reformulate Classical Mechanics as a timeless relativistic theory. Readers are introduced to a new class of reference systems, the binate frames, where physical events are identified with four position-coordinates -- no clocks are used.…
Formal definition of the reference frame is given. This definition is valid for nonrelativistic and relativistic cases. Proposed definition allows using wide classes of reference frames without restriction to inertial, uniformly accelerated…
It is shown that any dynamic equation on a configuration bundle $Q\to R$ of non-relativistic time-dependent mechanics is associated with connections on the affine jet bundle $J^1Q\to Q$ and on the tangent bundle $TQ\to Q$. As a consequence,…
The non-relativistic version of the multi-temporal quantization scheme of relativistic particles in a family of non-inertial frames (see hep-th/0502194) is defined. At the classical level the description of a family of non-rigid…
A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibrational) reference frames with the random initial conditions? One of the most general…
The usual formulations of time-dependent mechanics start from a given splitting $Y=R\times M$ of the coordinate bundle $Y\to R$. From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a…
The usual formulation of time-dependent mechanics implies a given splitting $Y=R\times M$ of an event space $Y$. This splitting, however, is broken by any time-dependent transformation, including transformations between inertial frames. The…
We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the…
Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is relational. This means that the configuration of an $N$-particle system is a shape, which is what remains when the effects of rotations,…
In this note we present invariant formulation of the d'Alambert principle and classical time-dependent Lagrangian mechanics with holonomic constraints from the perspective of moving frames.
We study the relativistic formulation of a classical time-dependent nonholonomic Lagrangian mechanics from the perspective of moving frames. We also introduce time-dependent $G$-Chaplygin systems with affine constraints, which are natural…
Newtonian physics is describes macro-objects sufficiently well, however it does not describe microobjects. A model of Extended Mechanics for Quantum Theory is based on an axiomatic generalization of Newtonian classical laws to arbitrary…
We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in…
We show that classical particle mechanics (Hamiltonian and Lagrangian consistent with relativistic electromagnetism) can be derived from three fundamental assumptions: infinite reducibility, deterministic and reversible evolution, and…
It is widely accepted that the fundamental geometrical law of nature should follow from an action principle. The particular subset of transformations of a system's dynamical variables that maintain the form of the action principle comprises…
Geometric (Schrodinger) quantization of nonrelativistic mechanics with respect to different reference frames is considered. In classical nonrelativistic mechanics, a reference frame is represented by a connection on a configuration space…